A spectral problem and an associated hierarchy of nonlinear evolution equations are presented in this article. In particular, the reductions of the two representative equations in this hierarchy are given: one is the nonlinear evolution equation rl= - ar,- 2icu/3] r2] r which looks like the nonlinear Schrijdinger equation, the other is the generalized derivative nonlinear Schrijdinger equation rt= $ar,,- ialr12r- a/3(lr12r),- a j3I r I 2r,-2iap21r14r which is just a combination of the nonlinear Schrijdinger equation and two different derivative nonlinear Schrodinger equations [D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 789 (1978); M. J. Ablowitz, A. Ramani, and H. Segur, J. Math. Phys. 21, 1006 (1980)]. The spectral problem is nonlinearized as a finite-dimensional completely integrable Hamiltonian system under a constraint between the potentials and the spectral functions. At the end of this article, the involutive solutions of the hierarchy of nonlinear evolution equations are obtained. Particularly, the involutive solutions of the reductions of the two representative equations are developed.
Qiao, Zhijun, "A hierarchy of nonlinear evolution equations and finite-dimensional involutive systems" (1994). Mathematical and Statistical Sciences Faculty Publications and Presentations. 12.
Journal of Mathematical Physics